Shuanhong Wang’s research while affiliated with Southeast University and other places

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Publications (110)


An algebraic framework for the Drinfeld double based on infinite groupoids
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  • File available

June 2023

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39 Reads

Nan Zhou

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Shuanhong Wang

The Drinfeld double associated to the weak multiplier Hopf (*-) algebra pairing A,B\left\langle A, B\right\rangle is constructed. We show that the Drinfeld double is again a weak multiplier Hopf (*-) algebra. If A and B are algebraic quantum groupoids, then so does the double. We also prove the correspondence between modules over the Drinfeld double and Yetter-Drinfeld modules. Finally, we prove that the double is a quasitriangular weak multiplier Hopf algebra.

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A Duality Theorem for Hopf Quasimodule Algebras

March 2023

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25 Reads

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1 Citation

Mathematics

In this paper, we introduce and study two smash products A★H for a left H-quasimodule algebra A over a Hopf quasigroup H over a field K and B#U for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our duality theorem (A★H)#H*≅A⊗(H#H*)≅A⊗Mn(K)≅Mn(A) in the setting of a Hopf quasigroup H of dimension n. As an application of our result, we consider a special case of a finite quasigroup.


A Morita-Takeuchi Context and Hopf Coquasigroup Galois Coextensions

February 2023

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19 Reads

Symmetry

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct C⋊H (as a symmetry of smash product) is linked to some quotient coalgebra Q=C/CH*+ by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for C/Q to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory.


Hopf Quasigroup Galois Extensions and a Morita Equivalence

January 2023

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34 Reads

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2 Citations

Mathematics

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product A#H is linked to the algebra of H invariants AH by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for A/AH to be Galois parallel in the case of H finite dimensional Hopf algebra.


Group-Graded By-Product Construction and Group Double Centralizer Properties

August 2022

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9 Reads

Mathematics

For a group π with unit e, we introduce and study the notion of a π-graded Hopf algebra. Then we introduce and construct a new braided monoidal category HHeYDπ over a π-graded Hopf algebra H. We introduce the notion of a π-double centralizer property and investigate this property by studying a braided π-graded Hopf algebra U(gln(V))⋉πH, where V is an n-dimensional vector space in HHeYDπ and U(gln(V)) is the braided universal enveloping algebra of gln(V) which is not the usual Hopf algebra. Finally, some examples and special cases are given.


Quasitriangular Hopf group-quasialgebras and generalized quantum Yang–Baxter equations

July 2022

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16 Reads

We introduce and study a large class of algebras (possibly nonassociative) with group-coalgebraic structures, Hopf nonassociative group-coalgebras, which provide a unifying framework for classical Hopf algebras, Turaev’s Hopf group-coalgebras, and Hopf quasigroups. Then, we introduce and discuss the notion of a quasitriangular Hopf π-quasialgebra and show that the category Qrep( H) R of quasirepresentations over any quasitriangular Hopf π-quasialgebra ( H, R) is a braided π-category. As an application of our theory, we give a new solution to the generalized quantum Yang–Baxter equation.


Double crossed biproducts and related structures

May 2022

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48 Reads

Let H be a bialgebra. Let σ:HHA\sigma: H\otimes H\to A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action \triangleright. Let τ:HHB\tau: H\otimes H\to B be a linear map, where B is a right H-comodule coalgebra, and an algebra with a right H-weak action \triangleleft. In this paper, we improve the necessary conditions for the two-sided crossed product algebra A#σH τ#BA\#^{\sigma} H~{^{\tau}\#} B and the two-sided smash coproduct coalgebra A×H×BA\times H\times B to form a bialgebra (called double crossed biproduct) such that the condition b[1]a0b[0]a1=abb_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b in Majid's double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzez\'nski's crossed product and give some applications.


Double crossed biproducts and related structures

April 2022

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21 Reads

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1 Citation

Communications in Algebra

Let H be a bialgebra. Let σ:H⊗H→A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action ⊳. Let τ:H⊗H→B be a linear map, where B is a right H-comodule coalgebra, and an algebra with a right H-weak action ⊲. In this paper, we improve the necessary conditions for the two-sided crossed product algebra A#σH τ#B and the two-sided smash coproduct coalgebra A×H×B to form a bialgebra (called double crossed biproduct) such that the condition b[1]⊳a0⊗b[0]⊲a−1=a⊗b in Majid’s double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzezński’s crossed product and give some applications.


A New Approach to Braided T-Categories and Generalized Quantum Yang–Baxter Equations

March 2022

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10 Reads

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2 Citations

Mathematics

We introduce and study a large class of coalgebras (possibly (non)coassociative) with group-algebraic structures Hopf (non)coassociative group-algebras. Hopf (non)coassociative group-algebras provide a unifying framework for classical Hopf algebras and Hopf group-algebras and Hopf coquasigroups. We introduce and discuss the notion of a quasitriangular Hopf (non)coassociative π-algebra and show some of its prominent properties, e.g., antipode S is bijective. As an application of our theory, we construct a new braided T-category and give a new solution to the generalized quantum Yang–Baxter equation.


Rota-Baxter operators on Turaev's Hopf group (co)algebras I: Basic definitions and related algebraic structures

January 2022

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67 Reads

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6 Citations

Journal of Geometry and Physics

We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 and 4 are given. At last we prove that Rota-Baxter Poisson T-algebras can produce pre-Poisson T-algebras and Poisson T-algebras can be obtained from pre-Poisson T-algebras.


Citations (53)


... The following notion is different from the one in [15,16]. ...

Reference:

A Duality Theorem for Hopf Quasimodule Algebras
Hopf Quasigroup Galois Extensions and a Morita Equivalence

Mathematics

... This algebra structure is denoted by A >⊳ V ⊲< C and is called the twosided crossed product afforded by the maps R 1 , R 2 , R 3 , E. This result admits also a converse. Particular cases are the iterated twisted tensor products of algebras, the so called two-sided crossed product over a quasi-bialgebra from [2], [6], and also a recent construction from [8]. ...

Double crossed biproducts and related structures
  • Citing Article
  • April 2022

Communications in Algebra

... 5 We also mention the works devoted to the analogous notions defined for associative conformal algebras and associative pseudoalgebras. In Ref. 9 and in Ref. 13, the associative Yang-Baxter equation was defined for associative conformal algebras and associative pseudoalgebras, respectively. In Ref. 8, the so-called conformal S-equation for left-symmetric conformal algebras was written down and studied. ...

Infinitesimal H -pseudobialgebra and associative pseudo-Yang-Baxter equation
  • Citing Article
  • September 2021

Communications in Algebra

... It is have been proved that the categories of Yetter-Drinfeld modules over weak Hopf quasigroups [18] and group-cograded Hopf algebras are braided [6]. Then there has a natural question: if we endow the weak Hopf quasigroups with a group-cograded structure, is it has the similar results with the weak Hopf quasigroups and group-cograded Hopf algebras case? ...

Yetter–Drinfel’d modules over weak Hopf quasigroups
  • Citing Article
  • August 2021

Journal of Algebra and Its Applications

... More generally, a Hopf quasigroup (non-associative algebra) was introduced by Klim and Majid [5], whose dual notion is a Hopf coquasigroup, a particular case of the notion of unital coassociative bialgebra introduced in [6]. For more research on these aspects, see the papers [7][8][9]. For some basic and recent papers related to non-associative BCC-algebras and B-filters in this field, we refer to [10][11][12]. ...

Double centralizer properties related to (co)triangular Hopf coquasigroups
  • Citing Article
  • September 2020

Communications in Algebra

... These have motivated some initial moves toward a unification of these two topics, non-associative and non-coassociative Hopf algebras, since none of the non-associative objects or the non-coassociative objects proposed up to now have been able to maintain the self-duality of the Hopf algebra concept. More recently, there have been developments in topics related to these Hopf algebras (see [16][17][18][19][20][21]). ...

Hopf quasicomodules and Yetter-Drinfel’d quasicomodules
  • Citing Article
  • July 2019

Communications in Algebra

... Later, Yau [17] [18] [19] [21] proposed the concepts of Hom-(co)modules, Hom-Hopf modules, and the duality of Hom-algebras, thus enriching the field of Hom-algebra theory. As an emerging field within algebra, (Bi)Hom-algebra is intimately connected to category theory (See,e.g., [22] [14]), combinatorics (See,e.g., [12]), braided group representations (See,e.g., [9]), and quantum algebras (See,e.g., [13]), producing a wealth of research outcomes in recent years. ...

A Braided T -Category Over Weak Monoidal Hom-Hopf Algebras
  • Citing Article
  • July 2019

Journal of Algebra and Its Applications